GroupTheory
IdentifySmallGroup
find where a group is in the small groups database
Calling Sequence
Parameters
Options
Description
Examples
Compatibility
IdentifySmallGroup(G, opts)
G
-
a group
opts
(optional) equations of the form keyword = value, listed below
assign = name
If given the option assign = x, where x is any name, IdentifySmallGroup will assign the isomorphism mapping G to H to the name x. This isomorphism can be used in the same way as the isomorphisms assigned by AreIsomorphic.
If x already has a value, then it needs to be protected from evaluation using quotation marks.
form = fpgroup or form = permgroup
This option can be used together with the assign option, explained above, in order to specify the form of the group H that is the codomain of the isomorphism to be assigned to the name specified in the assign option.
Specifying form = fpgroup results in the codomain being a finitely presented group. Specifying form = permgroup (the default) results in the codomain being a permutation group. You can equivalently specify the string forms of these values, as form = "fpgroup" or form = "permgroup".
If no assign option is specified, then the form option is ignored.
The command IdentifySmallGroup finds if a group H isomorphic to G occurs in the small groups database. (Currently, that means that the order of the group is at most 511.) If so, it returns the numbers under which H occurs in the database.
The value returned is a sequence of two numbers such that calling SmallGroup with those two numbers as arguments returns the group H. The first number is the order of G.
with⁡GroupTheory:
We identify the three-dimensional projective special linear group over the field of two elements.
IdentifySmallGroup⁡PSL⁡3,2
168,42
IdentifySmallGroup⁡PSL⁡2,7
We see that both groups are isomorphic (because they are both isomorphic to SmallGroup⁡168,42). Now construct a group using the SmallGroup command, then create a Cayley table group that is isomorphic to it, and test that it is still recognized as the same group.
g1≔SmallGroup⁡96,7
g1≔ < a permutation group on 96 letters with 6 generators >
g2≔CayleyTableGroup⁡g1
g2≔ < a Cayley table group with 96 elements >
IdentifySmallGroup⁡g2,assign=iso
96,7
Domain⁡iso
< a Cayley table group with 96 elements >
Codomain⁡iso
< a permutation group on 96 letters with 6 generators >
Using the infolevel facility, we can obtain some information about the progress of the command.
infolevelGroupTheory≔3
g3≔SmallGroup⁡128,1607
g3≔ < a permutation group on 128 letters with 7 generators >
IdentifySmallGroup⁡g3
IdentifySmallGroup: determined group size to be 128 IdentifySmallGroup: after evaluating hash value, candidates are {[128, 459, [a1, a2, a3, a4, a5, a6, a7], [[a1, a1, 1/a5], [1/a2, 1/a1, a2, a1, 1/a4], [1/a3, 1/a1, a3, a1], [1/a4, 1/a1, a4, a1], [1/a5, 1/a1, a5, a1], [1/a6, 1/a1, a6, a1], [1/a7, 1/a1, a7, a1], [a2, a2, 1/a6], [1/a3, 1/a2, a3, a2], [1/a4, 1/a2, a4, a2], [1/a5, 1/a2, a5, a2], [1/a6, 1/a2, a6, a2], [1/a7, 1/a2, a7, a2], [a3, a3], [1/a4, 1/a3, a4, a3], [1/a5, 1/a3, a5, a3], [1/a6, 1/a3, a6, a3], [1/a7, 1/a3, a7, a3], [a4, a4], [1/a5, 1/a4, a5, a4], [1/a6, 1/a4, a6, a4], [1/a7, 1/a4, a7, a4], [a5, a5, 1/a7], [1/a6, 1/a5, a6, a5], [1/a7, 1/a5, a7, a5], [a6, a6], [1/a7, 1/a6, a7, a6], [a7, a7]]], [128, 480, [a1, a2, a3, a4, a5, a6, a7], [[a1, a1, 1/a5], [1/a2, 1/a1, a2, a1, 1/a4], [1/a3, 1/a1, a3, a1], [1/a4, 1/a1, a4, a1], [1/a5, 1/a1, a5, a1], [1/a6, 1/a1, a6, a1], [1/a7, 1/a1, a7, a1], [a2, a2, 1/a6], [1/a3, 1/a2, a3, a2], [1/a4, 1/a2, a4, a2], [1/a5, 1/a2, a5, a2], [1/a6, 1/a2, a6, a2], [1/a7, 1/a2, a7, a2], [a3, a3 , 1/a5], [1/a4, 1/a3, a4, a3], [1/a5, 1/a3, a5, a3], [1/a6, 1/a3, a6, a3], [1/a7, 1/a3, a7, a3], [a4, a4], [1/a5, 1/a4, a5, a4], [1/a6, 1/a4, a6, a4], [1/a7, 1/a4, a7, a4], [a5, a5], [1/a6, 1/a5, a6, a5], [1/a7, 1/a5, a7, a5], [a6, a6, 1/a7], [1/a7, 1/a6, a7, a6], [a7, a7]]], [128, 483, [a1, a2, a3, a4, a5, a6, a7], [[a1, a1, 1/a5], [1/a2, 1/a1, a2, a1, 1/a4], [1/a3, 1/a1, a3, a1], [1/a4, 1/a1, a4, a1], [1/a5, 1/a1, a5, a1], [1/a6, 1/a1, a6, a1], [1/a7, 1/a1, a7, a1], [a2, a2, 1/a6], [1/a3, 1/a2, a3, a2], [1/a4, 1/a2, a4, a2], [1/a5, 1/a2, a5, a2], [1/a6, 1/a2, a6, a2], [1/a7, 1/a2, a7, a2], [a3, a3, 1/a5], [1/a4, 1/a3, a4, a3], [1/a5, 1/a3, a5, a3], [1/a6, 1/a3, a6, a3], [1/a7, 1/a3, a7, a3], [a4, a4], [1/a5, 1/a4, a5, a4], [1/a6, 1/a4, a6, a4], [1/a7, 1/a4, a7, a4], [a5, a5, 1/a7], [1/a6, 1/a5, a6, a5], [1/a7, 1/a5, a7, a5], [a6, a6], [1/a7, 1/a6, a7, a6], [a7, a7]]], [128, 1602, [a1, a2, a3, a4, a5, a6, a7], [[a1, a1, 1/a5], [1/a2, 1/a1, a2, a1, 1/a7], [1/a3, 1/a1, a3, a1], [1/a4, 1/a1, a4, a1], [1/a5, 1/a1, a5, a1], [1/a6, 1/a1, a6, a1], [1/a7, 1/a1, a7, a1], [a2, a2, 1/a6], [1/a3, 1/a2, a3, a2], [1/a4, 1/a2, a4, a2], [1/a5, 1/a2, a5, a2], [1/a6, 1/a2, a6, a2], [1/a7, 1/a2, a7, a2], [a3, a3], [1/a4, 1/a3, a4, a3], [1/a5, 1/a3, a5, a3], [1/a6, 1/a3, a6, a3], [1/a7, 1/a3, a7, a3], [a4, a4], [1/a5, 1/a4, a5, a4], [1/a6, 1/a4, a6, a4], [1/a7, 1/a4, a7, a4], [a5, a5, 1/a7], [1/a6, 1/a5, a6, a5], [1/a7, 1/a5, a7, a5], [a6, a6], [1/a7, 1/a6, a7, a6], [a7, a7]]], [128, 1603, [a1, a2, a3, a4, a5, a6, a7], [[a1, a1, 1/a5], [1/a2, 1/a1, a2, a1], [1/a3, 1/a1, a3, a1, 1/a7], [1/a4, 1/a1, a4, a1], [1/a5, 1/a1, a5, a1], [1/a6, 1/a1, a6, a1], [1/a7, 1/a1, a7, a1], [a2, a2, 1/a6], [1/a3, 1/a2, a3, a2], [1/a4, 1/a2, a4, a2], [1/a5, 1/a2, a5, a2], [1/a6, 1/a2, a6, a2], [1/a7, 1/a2, a7, a2], [a3, a3], [1/a4, 1/a3, a4, a3], [1/a5, 1/a3, a5, a3], [1/a6, 1/a3, a6, a3], [1/a7, 1/a3, a7, a3], [a4, a4], [1/a5, 1/a4, a5, a4], [1/a6, 1/a4, a6, a4], [1/a7, 1/a4, a7, a4], [a5, a5, 1/a7], [1/a6, 1/a5, a6, a5], [1/a7, 1/a5, a7, a5], [a6, a6], [1/a7, 1/a6, a7, a6], [a7, a7]]], [128, 1604, [a1, a2, a3, a4, a5, a6, a7], [[a1, a1, 1/a5], [1/a2, 1/a1, a2, a1], [1/a3, 1/a1, a3, a1], [1/a4, 1/a1, a4, a1], [1/a5, 1/a1, a5, a1], [1/a6, 1/a1, a6, a1], [1/a7, 1/a1, a7, a1], [a2, a2, 1/a6], [1/a3, 1/a2, a3, a2, 1/a7], [1/a4, 1/a2, a4, a2], [1/a5, 1/a2, a5, a2], [1/a6, 1/a2, a6, a2], [1/a7, 1/a2, a7, a2], [a3, a3], [1/a4, 1/a3, a4, a3], [1/a5, 1/a3, a5, a3], [1/a6, 1/a3, a6, a3], [1/a7, 1/a3, a7, a3], [a4, a4], [1/a5, 1/a4, a5, a4], [1/a6, 1/a4, a6, a4], [1/a7, 1/a4, a7, a4], [a5, a5, 1/a7], [1/a6, 1/a5, a6, a5], [1/a7, 1/a5, a7, a5], [a6, a6], [1/a7, 1/a6, a7, a6], [a7, a7]]], [128, 1605, [a1, a2, a3, a4, a5, a6, a7], [[a1, a1, 1/a5], [1/a2, 1/a1, a2, a1], [1/a3, 1/a1, a3, a1], [1/a4, 1/a1, a4, a1, 1/a7], [1/a5, 1/a1, a5, a1], [1/a6, 1/a1, a6, a1], [1/a7, 1/a1, a7, a1], [a2, a2, 1/a6], [1/a3, 1/a2, a3, a2, 1/a7], [1/a4, 1/a2, a4, a2], [1/a5, 1/a2, a5, a2], [1/a6, 1/a2, a6, a2], [1 /a7, 1/a2, a7, a2], [a3, a3], [1/a4, 1/a3, a4, a3], [1/a5, 1/a3, a5, a3], [1/a6, 1/a3, a6, a3], [1/a7, 1/a3, a7, a3], [a4, a4], [1/a5, 1/a4, a5, a4], [1/a6, 1/a4, a6, a4], [1/a7, 1/a4, a7, a4], [a5, a5, 1/a7], [1/a6, 1/a5, a6, a5], [1/a7, 1/a5, a7, a5], [a6, a6], [1/a7, 1/a6, a7, a6], [a7, a7]]], [128, 1606, [a1, a2, a3, a4, a5, a6, a7], [[a1, a1, 1/a5], [1/a2, 1/a1, a2, a1], [1/a3, 1/a1, a3, a1], [1/a4, 1/a1, a4, a1], [1/a5, 1/a1, a5, a1], [1/a6, 1/a1, a6, a1], [1/a7, 1/a1, a7, a1], [a2, a2, 1/a6], [1/a3, 1/a2, a3, a2], [1/a4, 1/a2, a4, a2], [1/a5, 1/a2, a5, a2], [1/a6, 1/a2, a6, a2], [1/a7, 1/a2, a7, a2], [a3, a3], [1/a4, 1/a3, a4, a3, 1/a7], [1/a5, 1/a3, a5, a3], [1/a6, 1/a3, a6, a3], [1/a7, 1/a3, a7, a3], [a4, a4], [1/a5, 1/a4, a5, a4], [1/a6, 1/a4, a6, a4], [1/a7, 1/a4, a7, a4], [a5, a5, 1/a7], [1/a6, 1/a5, a6, a5], [1/a7, 1/a5, a7, a5], [a6, a6], [1/a7, 1/a6, a7, a6], [a7, a7]]], [128, 1607, [a1, a2, a3, a4, a5, a6, a7], [[a1, a1, 1/a5], [1/a2, 1/a1, a2, a1, 1/a7], [1/a3, 1/a1, a3, a1], [1/a4, 1/a1, a4, a1], [1/a5, 1/a1, a5, a1], [1/a6, 1/a1, a6, a1], [1/a7, 1/a1, a7, a1], [a2, a2, 1/a6], [1/a3, 1/a2, a3, a2], [1/a4, 1/a2, a4, a2], [1/a5, 1/a2, a5, a2], [1/a6, 1/a2, a6, a2], [1/a7, 1/a2, a7, a2], [a3, a3], [1/a4, 1/a3, a4, a3, 1/a7], [1/a5, 1/a3, a5, a3], [1/a6, 1/a3, a6, a3], [1/a7, 1/a3, a7, a3], [a4, a4], [1/a5, 1/a4, a5, a4], [1/a6, 1/a4, a6, a4], [1/a7, 1/a4, a7, a4], [a5, a5, 1/a7], [1/a6, 1/a5, a6, a5], [1/a7, 1/a5, a7, a5], [a6, a6], [1/a7, 1/a6, a7, a6], [a7, a7]]], [128, 1634, [a1, a2, a3, a4, a5, a6, a7], [[a1, a1, 1/a6], [1/a2, 1/a1, a2, a1, 1/a5], [1/a3, 1/a1, a3, a1], [1/a4, 1/a1, a4, a1], [1/a5, 1/a1, a5, a1], [1/a6, 1/a1, a6, a1], [1/a7, 1/a1, a7, a1], [a2, a2, 1/a5], [1/a3, 1/a2, a3, a2], [1/a4, 1/a2, a4, a2], [1/a5, 1/a2, a5, a2], [1/a6, 1/a2, a6, a2], [1/a7, 1/a2, a7, a2], [a3, a3], [1/a4, 1/a3, a4, a3], [1/a5, 1/a3, a5, a3], [1/a6, 1/a3, a6, a3], [1/a7, 1/a3, a7, a3], [a4, a4], [1/a5, 1/a4, a5, a4], [1/a6, 1/a4, a6, a4], [1/a7, 1/a4, a7, a4], [a5, a5], [1/a6, 1/a5, a6, a5], [1/a7, 1/a5, a7, a5], [a6, a6, 1/a7], [1/a7, 1/a6, a7, a6], [a7, a7]]], [128, 1649, [a1, a2, a3, a4, a5, a6, a7], [[a1, a1, 1/a6], [1/a2, 1/a1, a2, a1, 1/a5], [1/a3, 1/a1, a3, a1], [1/a4, 1/a1, a4, a1], [1/a5, 1/a1, a5, a1], [1/a6, 1/a1, a6, a1], [1/a7, 1/a1, a7, a1], [a2, a2], [1/a3, 1/a2, a3, a2], [1/a4, 1/a2, a4, a2], [1/a5, 1/a2, a5, a2], [1/a6, 1/a2, a6, a2], [1/a7, 1/a2, a7, a2], [a3, a3, 1/a5], [1/a4, 1/a3, a4, a3], [1/a5, 1/a3, a5, a3], [1/a6, 1/a3, a6, a3], [1/a7, 1/a3, a7, a3], [a4, a4], [1/a5, 1/a4, a5, a4], [1/a6, 1/a4, a6, a4], [1/a7, 1/a4, a7, a4], [a5, a5], [1/a6, 1/a5, a6, a5], [1/a7, 1/a5, a7, a5], [a6, a6, 1/a7], [1/a7, 1/a6, a7, a6], [a7, a7]]], [128, 1696, [a1, a2, a3, a4, a5, a6, a7], [[a1, a1, 1/a6], [1/a2, 1/a1, a2, a1, 1/a5], [1/a3, 1/a1, a3, a1], [1/a4, 1/a1, a4, a1], [1/a5, 1/a1, a5, a1], [1/a6, 1/a1, a6, a1], [1/a7, 1/a1, a7, a1], [a2, a2], [1/a3, 1/a2, a3, a2], [1/a4, 1/a2, a4, a2], [1/a5, 1/a2, a5, a2], [1/a6, 1/a2, a6 , a2], [1/a7, 1/a2, a7, a2], [a3, a3, 1/a6], [1/a4, 1/a3, a4, a3], [1/a5, 1/a3, a5, a3], [1/a6, 1/a3, a6, a3], [1/a7, 1/a3, a7, a3], [a4, a4, 1/a5], [1/a5, 1/a4, a5, a4], [1/a6, 1/a4, a6, a4], [1/a7, 1/a4, a7, a4], [a5, a5], [1/a6, 1/a5, a6, a5], [1/a7, 1/a5, a7, a5], [a6, a6, 1/a7], [1/a7, 1/a6, a7, a6], [a7, a7]]], [128, 1720, [a1, a2, a3, a4, a5, a6, a7], [[a1, a1, 1/a6], [1/a2, 1/a1, a2, a1, 1/a5], [1/a3, 1/a1, a3, a1], [1/a4, 1/a1, a4, a1, 1/a5], [1/a5, 1/a1, a5, a1], [1/a6, 1/a1, a6, a1], [1/a7, 1/a1, a7, a1], [a2, a2], [1/a3, 1/a2, a3, a2, 1/a5], [1/a4, 1/a2, a4, a2], [1/a5, 1/a2, a5, a2], [1/a6, 1/a2, a6, a2], [1/a7, 1/a2, a7, a2], [a3, a3], [1/a4, 1/a3, a4, a3], [1/a5, 1/a3, a5, a3], [1/a6, 1/a3, a6, a3], [1/a7, 1/a3, a7, a3], [a4, a4, 1/a5], [1/a5, 1/a4, a5, a4], [1/a6, 1/a4, a6, a4], [1/a7, 1/a4, a7, a4], [a5, a5], [1/a6, 1/a5, a6, a5], [1/a7, 1/a5, a7, a5], [a6, a6, 1/a7], [1/a7, 1/a6, a7, a6], [a7, a7]]]} IdentifySmallGroup: testing possible id 459 IdentifySmallGroup: testing possible id 480 IdentifySmallGroup: testing possible id 483 IdentifySmallGroup: testing possible id 1602 IdentifySmallGroup: testing possible id 1603 IdentifySmallGroup: testing possible id 1604 IdentifySmallGroup: testing possible id 1605 IdentifySmallGroup: testing possible id 1606 IdentifySmallGroup: testing possible id 1607
128,1607
The GroupTheory[IdentifySmallGroup] command was introduced in Maple 17.
For more information on Maple 17 changes, see Updates in Maple 17.
See Also
GroupTheory[AllSmallGroups]
GroupTheory[SmallGroup]
Download Help Document
What kind of issue would you like to report? (Optional)